Can any relation in the group be derived from the relations in the presentation?

Question

I am looking at page 26 of Dummit and Foote, and I see the following statement discussing the presentation of $D_{2n} = \langle r, s\mid r^2=1, s^n=1, rs=sr^{-1} \rangle$.

...$D_{2n}$ has the relations $r^2=1, s^n=1, rs=sr^{-1}$. Moreover, these relations have the property that any other relation among the elements of $S = \{r, s \}$ can be deduced from these three.

My question is, is this true for all group presentations? That is, can any relation between elements of the generators be determined from the relations in the presentation?

My inclination is no; because a few lines below, Dummit and Foote say

...in an arbitrary presentation it may be extremely difficult (or even impossible) to tell when two elements of the group (expressed in terms of the given generators) are equal.

Answer 1

Can any relation between elements of the generators be determined from the relations in the presentation?

Yes! Sometimes, a group is defined by one of its presentations, so there's no other choice when that happens.

My inclination is no; because a few lines below, Dummit and Foote say

...in an arbitrary presentation it may be extremely difficult (or even impossible) to tell when two elements of the group (expressed in terms of the given generators) are equal.

This is known as a decidability result. It's the word problem. Generally speaking, given two elements of a group given by a presentation, it is literally impossible to decide whether or not one is equal to another using a Turing machine in finite time.

But this does not mean that any and all relations between elements of a group cannot be determined by the relations of a presentation of that group.

Answer 2

It's all about the difference between "being mathematically determined" and "being computable". All relations between elements are logically obtainable from the relations in the presentation, but it is not a computable problem in general (of course for some groups it is, like for finite groups, since at worst you can just write the whole multiplication table).